This section is intended to introduce various aspects of the art, which may be associated with exemplary embodiments of the present techniques. This discussion is believed to assist in providing a framework to facilitate a better understanding of particular aspects of the present techniques. Accordingly, it should be understood that this section should be read in this light, and not necessarily as admissions of prior art.
Modern society is greatly dependant on the use of hydrocarbons for fuels and chemical feedstocks. Hydrocarbons are generally found in subsurface rock formations that are generally termed reservoirs. Removing hydrocarbons from the reservoirs depends on numerous physical properties of the rock formations, such as the permeability of the rock containing the hydrocarbons, the ability of the hydrocarbons to flow through the rock formations, and the proportion of hydrocarbons present, among others.
Often, mathematical models termed “simulation models” are used to simulate hydrocarbon reservoirs for locating hydrocarbons and optimizing the production of the hydrocarbons. A simulation model is a type of computational fluid dynamics simulation where a set of partial differential equations (PDE's) which govern multi-phase, multi-component fluid flow through porous media and the connected facility network can be approximated and solved. This is an iterative, time-stepping process where a particular hydrocarbon production strategy is optimized.
Properties for reservoir simulation models, such as permeability or porosity, are often highly heterogeneous and can vary. The variation is at all length scales from the smallest to the largest scales that can be comparable to the reservoir size. Computer simulations of such models that use a very fine grid discretization to capture the heterogeneity are computationally very expensive. However, disregarding the heterogeneity can lead to wrong results.
In order to achieve reasonable computational performance, the reservoir properties are often upscaled. For example, a homogenization technique may be employed to define the reservoir properties on coarser simulation grids. This technique in different variants has been widely used in the academia and the industry with reasonable success. Nevertheless, upscaling may have numerous drawbacks. As an example, upscaling may not work well for problems with non-separable scales, as discussed in greater detail below. Further, upscaling may not fully capture global features in a reservoir flow. Upscaling may also have difficulties estimating errors when complicated flows are modeled.
Heterogeneous or multiscale phenomena can be classified in two categories: separable scales and non-separable scales. It is common for a porous media to have multiple scales on which the properties of the media vary. The scales in reservoir may be not separable, for example, if the features vary continuously across the reservoir. Upscaling methods may resolve models with separable scales, but may not be able to correctly resolve non-separable models.
In addition to non-separable phenomena, there are features like channels, long fractures and faults that can extend through a large part of the reservoir. These may be referred to as global features and information. Current upscaling methods can miss the influence of the global features that could be significant to flow simulations, i.e., there may not be information that upscaling algorithms can use to generate more complicated models. Thus, the results may not be very realistic.
Another approach that may be used instead of, or in addition to, upscaling is multiscale simulation. In a multiscale simulation, the computations are still performed on a coarser grid, but fine grid information is used to construct a set of basis functions that may be used for mapping fine grid properties to the coarse grid. Multiscale simulations can be orders of magnitude faster than simulations on the fine grid, providing solutions that are of comparable quality. One multiscale method is the multiscale finite element method (MsFEM). See, e.g., T. Y. Hou and X. H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134:169-189, 1997. This technique shares similarities to other numerical multiscale methods, such as variational multiscale finite element methods. See T. Arbogast and K. Boyd, Subgrid upscaling and mixed multiscale finite elements, SIAM Num. Anal., 44:1150-1171, 2006; see also T. J. R. Hughes, G. R. Feijoo, L. Mazzei, and J.-B. Quincy, The variational multiscale method—A paradigm for computational mechanics, Comput. Meth. Appl. Mech. Eng., 166:3-24, 1998. The techniques may also be similar to heterogeneous multiscale methods proposed by W. E and B. Engquist, The heterogeneous multi-scale methods, Comm. Math. Sci., 1(1) (2003), pp. 87-133. Another approach that has been developed is a mixed multiscale finite element method MMsFEM, which is locally mass conservative and can be applied to multiphase simulations. See Z. Chen and T. Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillatory coefficients, Math. Comp., 72:541-576, 2002. As these references show, intensive research into the use of multiscale methods has been performed for more than ten years, especially in academia. The simulation problems solved with multiscale methods have been simplified and are more academic than practical, though there is a trend to apply multiscale methods to more realistic and complex problems.
The multiscale methods discussed above use local information and perform well for separable scales. However, multiscale methods using only local information may suffer from resonance errors, which can be the dominant error in multiscale simulations. Resonance errors show up as an oscillation of a function across a simulation cell, increasing in magnitude near the edges of a cell in a simulation mesh. The resonance errors are usually proportional to the ratio between the characteristic length scale and the coarse mesh size. The ratios are small when the magnitudes of the characteristic length scales are distinctly different from the magnitude of the coarse mesh size. The ratios become large when the characteristic length scales are close to the coarse mesh size. Thus, the errors show up for the entire range of the characteristic length scales, and the errors are different for different characteristic length scales. See, e.g., Y. Efendiev, V. Gunting, T. Y. Hou, and R. Ewing, Accurate multiscale finite element methods for two-phase flow simulations, J. Comp. Phys., 220(1):155-174, 2006. Using some limited global information may be useful for the construction of basis functions in order to develop multiscale methods that reduce or remove resonance errors, see J. E. Aarnes, Y. Efendiev, and L. Jiang, Mixed multiscale finite element methods using limited global information, SIAM MMS, 7(2):655-676, 2008. Moreover, MsFEMs using global information may be applicable in problems without scale separation.
Industrial and academic researchers have reported results for a similar technique, the MultiScale Finite Volume method (MSFV). See e.g. P. Jenny, S. H. Lee, and H. A. Tchelepi, Multi-scale finite-volume method for elliptic problems in subsurface flow simulation, J. Comput. Phys., 187 (2003), pp. 47-67; see also P. Jenny, S. H. Lee, and H. A. Tchelepi, Adaptive multiscale finite-volume method for multiphase flow and transport, SIAM MMS, 3 (2004), pp. 50-64.
For example, U.S. Pat. No. 7,496,488 to Jenny, et al., discloses a multi-scale finite-volume method for use in subsurface flow simulation. In the method, a multi-scale finite-volume (MSFV) method is used to solve elliptic problems with a plurality of spatial scales arising from single or multi-phase flows in porous media. The method efficiently captures the effects of small scales on a coarse grid, is conservative, and treats tensor permeabilities correctly. The underlying idea is to construct transmissibilities that capture the local properties of a differential operator. This leads to a multi-point discretization scheme for a finite-volume solution algorithm. Transmissibilities for the MSFV method are preferably constructed only once as a preprocessing step and can be computed locally.
U.S. Patent Application Publication No. 2008/0208539 by Lee, et al., discloses a method, apparatus and system for reservoir simulation using a multi-scale finite volume method including black oil modeling. The multi-scale finite-volume (MSFV) method simulates nonlinear immiscible three-phase compressible flow in the presence of gravity and capillary forces. Consistent with the MSFV framework, flow and transport are treated separately and differently using a fully implicit sequential algorithm. The pressure field is solved using an operator splitting algorithm. The general solution of the pressure is decomposed into an elliptic part, a buoyancy/capillary force dominant part, and an inhomogeneous part with source/sink and accumulation. A MSFV method is used to compute the basis functions of the elliptic component, capturing long range interactions in the pressure field. Direct construction of the velocity field and solution of the transport problem on the primal coarse grid provides flexibility in accommodating physical mechanisms. The MSFV method computes an approximate pressure field, including a solution of a course-scale pressure equation; constructs fine-scale fluxes; and computes a phase-transport equation.
As described in the references above, the MSFV methods have been used in structured Cartesian grids and are closely related to the Multi-Point Flux Approximation schemes (MPFA) popular in the oil industry. MSFV relies on two sets of basis functions, both for the coarse-scale pressure.
However, there are several problems with the existing approaches for finite volume discretizations using the MSFV. For example, solving problems for the basis function can be computationally expensive. Further, the MSFV has not been extended to unstructured grids and global information is not used in the MSFV. Geometric information can be needed to impose boundary conditions, which is very difficult to implement for unstructured grids. In addition, the multiscale basis uses only local information and cannot resolve global features like channels, impermeable shale barriers, fractures, etc.